An observation algorithm for nonlinear systems with unknown inputs

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An observation algorithm for nonlinear systems with unknown inputs

Jean-Pierre Barbot, Driss Boutat, Thierry Floquet

To cite this version:

Jean-Pierre Barbot, Driss Boutat, Thierry Floquet. An observation algorithm for non- linear systems with unknown inputs. Automatica, Elsevier, 2009, 45 (8), pp.1970-1974.

�10.1016/j.automatica.2009.04.009�. �inria-00391819�

An observation algorithm for nonlinear systems with unknown inputs

Jean-Pierre Barbot

, Driss Boutat

and T. Floquet

1

Equipe Commande des Syst`emes (ECS), ENSEA, 6 Avenue du Ponceau, 95014 Cergy, France.

2

LVR/ENSIB, 10 Boulevard de Lahitolle, 18020 Bourges, France.

3

LAGIS UMR CNRS 8146, Ecole Centrale de Lille, BP 48, Cit´e Scientifique, 59651 Villeneuve-d’Ascq, France

4

Equipe Projet ALIEN, INRIA Lille - Nord Europe, France.

Abstract

This paper provides some new developments in the design of unknown input observers for nonlinear systems. An algorithm which states if the state and the unknown input of the system can be recovered in finite time is introduced. This algorithm leads to the transformation of the system into an extended block triangular observable form suitable for the design of finite time observers. The proposed method is useful to relax some restrictive conditions of existing nonlinear unknown input observer design procedures.

1 Introduction

It is of importance to design observers for multivariable linear or nonlinear sys- tems partially driven by unknown inputs. Such a problem arises in systems subject to disturbances or with inaccessible inputs and in many applications such as parameter identification, fault detection and isolation or cryptography.

The design of observers for nonlinear systems is a challenging problem, even for accurately known systems. It has received a considerable amount of attention in the literature. In many approaches, nonlinear coordinate transformations are used to transform the system into suitable observer canonical forms. Then, observers with linearizable error dynamics (see e.g. [9, 20, 21]), high gain ob- servers [8] or backstepping observers [15] can be designed. Few works deal with the design of unknown input observers for nonlinear systems. Some of them are concerned with applications in the field of fault detection and identification, and in particular the nonlinear Fundamental Problem of Residual Generation [4, 10, 19]. Other ones deal with nonlinear systems subject to exogenous per- turbations and are based on sliding mode considerations [22]. The convergence is obtained in finite time under the assumptions that the system can be put

into a set of triangular observable forms, where the unknown inputs act only on the last dynamics of each triangular form. This assumption is known as the observability matching condition. The present work aims at the development of a systematic method leading to the finite time observation of a class of nonlin- ear systems with unknown inputs even if the observability matching condition is not fulfilled. To this end, the procedure given in [5] is extended to the nonlin- ear case. It also results in a constructive algorithm that transforms the system into a similar type of block triangular observable forms. This transformation relies on the introduction of suitable fictitious outputs. Sufficient conditions for the existence of such auxiliary variables are given. After transformation of the system, it is shown that it is possible to design observers that provide the finite time estimation of both the state variables and the unknown inputs. An illustrative example highlights the efficiency of the proposed methodology.

2 Problem statement and motivations

Consider, on an open setU, the nonlinear system:







˙

x=f(x) +g(x)w=f(x) +P^m

i=1

gi(x)wi

y=h(x) = [h1(x), .., hp(x)]^T

(1)

where x ∈ U ⊂ Rⁿ is the state vector, y ∈ R^p is the output vector and wherew= [w1, ..., wm]∈R^m represents the unknown inputs. The vector fields f and g1, ..., gm, and the functions h1, ..., hp, are assumed to be sufficiently smooth on U. Without loss of generality, it is assumed that p≥m, and that for all x ∈ U, the distribution G = span{g1, . . . , gm} and the codistribution span{dh1, . . . , dhp} are nonsingular on U. Like in [17], let us first define the unknown input characteristic indexes{ρ1, ..., ρp} such that, for 1≤i≤p:

LgjL^k_fhi(x) = 0, fork < ρi−1, and for all 1≤j≤m, LgjL^ρ_fⁱ⁻¹hi(x)6= 0, for at least one 1≤j ≤m,

for all x ∈ U. The system (1) with w = 0 is supposed to be locally weakly observable onU [11]. Thus, there exists a change of coordinates

φ=³

h1, ..., L^ν_f¹⁻¹h1, ..., hp, ..., L^ν_f^p−1hp

´T

such that, after a suitable reordering of the state components, the system (1) is locally transformed into

ξ˙i=Aδiξi+HδiVδi(x, w) (2)

η. =a(ξ, η) +b(ξ, η)w (3)

yi=Cδiξi

withξ=¡

ξ₁^T, ..., ξ^T_p¢T

, ξi∈R^δi, 1≤i≤p Vδi(x, w) =L^δ_fⁱhi(x) +P^m

i=1

LgjL^δ_fⁱ⁻¹hi(x)wj ∈R

Aδi =















0 1 0 0 0

0 0 1 0 0

... ... ... . .. ...

0 0 0 0 1

0 0 0 0 0















∈R^δi×δi

Hδi =¡

0 0 ... 1 ¢T

∈R^δi, Cδi =¡

1 0 ... 0 ¢

∈R^1×δi

and a, b are smooth vector fields on U. The integers (ν1, ν2, . . . , νp) are the so-called observability indices of system (1) (see [14] for a definition) and thus satisfyν1+· · ·+νp=n. From the definition of theρi, one hasδi= min (νi, ρi) andgδi = 0 ifνi< ρi (i.e. δi=νi).

2.1 A triangular observable form

Most existing nonlinear observers for system (1) are designed under the assump- tion that the system satisfies the so-called observability matching condition.

This condition was first formulated for SISO linear systems in [18], Chapter 4. In the general case of MIMO nonlinear systems, a necessary and sufficient condition is given by:

νi≤ρifor all 1≤i≤p. (4)

Then,δi =νi for all 1≤i≤p, and under the change of coordinatesξ =φ(x) the system (1) is transformed into the form:

ξ˙i=Aνiξi+HνiVνi(ξ, w) (5) yi=Cνiξi

with P^m

i=1

LgjL^ν_fⁱ⁻¹hi(x)wj6= 0 if and only ifνi =ρi. Each subsystem of (5) is in the so-called triangular observable form. Finite time observers for such a form can be found in the literature, based on step-by-step sliding mode techniques [22], higher order sliding modes [6, 16], numerical approaches [3, 13], or algebraic methods [2]. The design of such observers is left to the reader, since they are straightforward applications of existing results. Nevertheless, depending on the choice of the observer, some assumptions have to be introduced. For instance, in all the previously mentioned works, the unknown inputwhas, at least, to be bounded. A necessary and sufficient condition for the recovery of the unknown inputs is that

Γ(x) =









Lg1L^ρ_f¹⁻¹h1(x) ... LgmL^ρ_f¹⁻¹h1(x)

... ...

Lg1L^ρ_f^p−1hp(x) ... LgmL^ρ_f^p−1hp(x)









has rankm. In this case,ρ={ρ1, ..., ρp}is the vector relative degree as defined in [12], p. 220, whenm=p.

2.2 An extended triangular observable form

The aim of this paper is to provide an observation algorithm that allows for the finite time estimation of both the state and the unknown inputs of (1) even ifνj > ρj for at least aj in {1, ..., p}. Consider again the general form (2-3).

Applying any of the aforementioned finite time observers: (i) for 1 ≤ i ≤ p, ξi can be estimated in finite time; (ii) one can also recover in finite time the last component Vδi of each subsystem of (2). The problem is to recover the remaining stateη. Denote:

V(x) =











Vδ1(x, w) Vδ2(x, w)

... Vδp(x, w)











=













L^δ_f¹h1(x) L^δ_f²h2(x)

... L^δ_f^php(x)













+ Γδ(x)w

where

Γδ(x) =









Lg1L^δ_f¹⁻¹h1(x) ... LgmL^δ_f¹⁻¹h1(x)

... ...

Lg1L^δ_f^p−1hp(x) ... LgmL^δ_f^p−1hp(x)







 .

Let£be the commutative algebra of the measured outputs and their successive Lie derivatives up to orderδi: £= span{h1, ..., L^δ_f¹⁻¹h1, ..., hp, ..., L^δ_f^p−1hp}and letd£be the codistribution:

d£= span{dh1, ..., dL^δ_f¹⁻¹h1, ..., dhp, ..., dL^δ_f^p−1hp}.

Assume there exists a 1×prow vectorK(x) = (k1(x), ..., kp(x))6= 0,ki∈£for 1≤i≤p, such that:

K(x)Γδ(x) = 0 for allx∈U (6)

and set:

¯

y = ¯h(x) =K(x)V(x) =

p

X

i=1

ki(x)L^δ_fⁱhi(x) (7) Note that ¯y is an available information (after a finite time) and is not affected by the unknown inputs. Therefore, ifd£+span©

d¯hª

d£, ¯ycan be considered as an additional fictitious output. Then, let ¯ρi and ¯νi be the unknown input characteristic indexes and the observability indices of (1) with respect to the extended output £

y^T,y¯¤T

=£

h^T,¯h(x)¤T

. If ¯νi ≤ρ¯i for all 1≤i ≤p+ 1, the

system (1) can be transformed into:

ξ˙i=Aν¯iξi+Hν¯iVν¯i( ˜ξ, w) for 1≤i≤p yi=C¯νiξi

ξ˙¯=Aν¯p+1ξ¯+Hν¯p+1Vν¯p+1( ˜ξ, w)

¯

y=C¯νp+1ξ¯ where ˜ξ=¡

ξ₁^T, ..., ξ_p^T,ξ¯^T¢T

∈Rⁿ. Then, it is possible to recover both the state and the unknown inputs in finite time. Let us give sufficient conditions for the existence of a suitable fictitious output ¯y. For this, the following notations are introduced:

i) G^⊥ = span{̟1, ..., ̟n−m}, the annihilator of G (̟i are 1-forms such that ιgk̟i = 0, where ιg̟ is the inner product of the vector fieldg and the 1-form

̟).

ii) Ω^£, the module spanned byd£ over£.

Proposition 1 The following conditions are equivalent:

i) Equation (6) has a solutionK andKV /∈£.

ii) Ξ =span{̟∈G^⊥∩Ω^£ such that ιf̟ /∈£} 6={0}.

Proof: Set̟=

p

P

i=1

kidL^δ_fⁱ⁻¹hi withki∈£. Clearly,̟∈Ω^£and

ιf̟=ιf p

X

i=1

kidL^δ_fⁱ⁻¹hi=

p

X

i=1

kiL^δ_fⁱhi=KV = ¯y.

KΓδ =K









dL^δ_f¹⁻¹h1

... dL^δ_f^p−1hp









£ g1 · · · gm ¤

=̟.£

g1 · · · gm ¤

Thus: {K is a solution of (6) such that KV /∈£} ⇐⇒ {̟ ∈Ω£, ιτ̟ = 0 for anyτ∈Gandιf̟ /∈£} ⇐⇒ {̟∈G^⊥∩Ω^£ andιf̟ /∈£} ⇐⇒Ξ6={0}. ¤ The discussion above can be recursively generalized as follows. Assume that the condition (4) is not still satisfied with the extended output obtained with the solutions of (6). On the basis of this new output, the corresponding matrix Γδ¯ can be computed and another set of fictitious outputs can eventually be found. One can iterate this procedure until the condition (4) is fulfilled for a new extended output. Then, the original system can be put into an extended

block triangular observable form¹: ξ˙_i¹=A_ν¹

iξ_i¹+H_ν¹

iV_ν¹

i(ξ, w) y_i¹=yi=C_νi¹ξ_i¹, 1≤i≤p¹ ξ˙_i²=A_ν²

iξ_i²+H_ν²

iV_ν²

i(ξ, w) y_i²=C_ν²

iξ_i², 1≤i≤p² ...

ξ˙_i^k∗ =A_νk∗

i ξ_i^k∗+H_νk∗

i V_νk∗

i (ξ, w) y_i^k∗ =C_νk∗

i ξ_i^k∗, 1≤i≤p^k∗ (8)

where the integers ν_i^j are the observability indices of the system (1) with the new outputsy_i^j. The first subsystem is fed by the original outputs of the system.

A finite time observer is designed to estimate the state of this subsystem and to provide in finite time the knowledge of the fictitious outputsy²_i, 1≤i≤p². Then, the state of the second triangular observable form can be estimated as well as the fictitious outputs y_i³. Thus, one can recursively obtain the whole state of the system in finite time.

Remark 1 The finite time property is required to ensure that one obtains a fast and accurate estimation of the fictitious outputs (for instance, via the equivalent output injection in the case of sliding mode observers, see [6] and the references therein). Furthermore, this property is often desirable in the framework of ob- servation and particularly for the purpose of observer-based controller design for nonlinear systems. Then, for a large class of nonlinear systems, the observer can be designed separately from the controller and the separation principle does not need to be proved. It can also be of paramount importance in applications that require fast estimations of some unknown inputs like fault detection and identification or on-line parameter identification.

3 Nonlinear unknown input observer algorithm

An algorithm that states if the system can be transformed into (8) and that provides the integersp^j, ν_i^j and the auxiliary outputsy^j_i (j = 1, ..., k^∗) is now given.

Step 0: Compute G and its annihilatorG^⊥. Set p¹ = p, h

h¹₁, .., h¹_p1

iT

= [h1, .., hp]^T,µ⁰= 0,z₁⁰=...=z⁰_µ0 = 0.

Stepα: [a]Consider y^α=£

h^α₁, .., h^α_pα

¤T

∈R^pα and reorder its components as follows:

y^α=£

h^α₁, ..., h^α_lα, h^α_lα+1, ..., h^α_pα

¤T

1Systems that admit such a form belong to the class of left invertible systems with trivial zero dynamics (see [2]).

such that for 1≤j≤l^α:

∀i∈[1, . . . , m], ∀k∈N LgiL^k_fh^α_j = 0 and for 1≤j≤p^α−l^α, there exists an integerρ^α_j such that:

∀i ∈ [1, . . . , m] LgiL^k_fh^α_l^α_+j = 0 ∀k < ρ^α_j −1

∃i ∈ [1, . . . , m] LgiL^ρ

α j−1

f h^α_l^α_+j 6= 0.

[b]Define Φ^α={dh^α₁, ..., dLⁿ_f⁻¹h^α₁, ..., dh^α_lα, ..., dLⁿ_f⁻¹h^α_lα}. Compute I^α= spann³

dz^α₁⁻¹, ..., dz_µ^αα−⁻¹1

´ [Φ^αo . Let dimI^α=µ^α−1+ϕ^α. I^α can be written as follows:

I^α= span{dz₁^α−1, ..., dz_µ^αα−1⁻¹, dh^α₁, ., dL^ϕ_f^α1−1h^α₁, ., dh^α_lα, ., dL^ϕ_f^αlα−1h^α_lα} with ^l

α

P

i=1

ϕ^α_i =ϕ^α. Ifµ^α−1+ϕ^α=n, set µ^k∗ = µ^α−1+ϕ^α ndz₁^k∗, ..., dz_µ^k∗k∗

o = {dz^α₁⁻¹, ..., dz_µ^αα−⁻¹1, dh^α₁, ., dL^ϕ_f^α1−1h^α₁, ., dh^α_l^α, ., dL^ϕ_f^αlα−1h^α_l^α} and stop the algorithm.

[c]Ifµ^α−1+ϕ^α< n, consider the outputs affected by the unknown inputs and define:

Υ^α={dh^α_lα+1, ., dL^ρ_f^α1−1h^α_lα+1, ., dh^α_pα, ., dL^ρ

α pα−lα−1 f h^α_pα} Compute the codistribution Ω^α= span{I^αS

Υ^α}. Let dimΩ^α=µ^α−1+ϕ^α+ κ^α=µ^αand write Ω^α= span{dz₁^α, ..., dz^α_µα} with

{dz₁^α, ..., dz_µ^αα}={dz₁^α−1, ..., dz_µ^αα−1⁻¹, dh^α₁, ..., dL^ϕ_f^α1−1h^α₁, ..., dh^α_lα, ..., dL^ϕ_f^αlα−1h^α_lα, dh^α_lα+1, ..., dL^κ_f^α1−1h^α_lα+1, ..., dh^α_pα, ..., dL^κ

α pα−lα−1 f h^α_pα}.

and

p^α−l^α

P

i=1

κ^α_i =κ^α. Ifµ^α=n, the algorithm stops.

[d]Otherwise,µ^α< n. Define

£^α = span{z₁^α, ..., z_µ^αα} Ω^α£ = span

(_µ^α X

i=1

φidz_i^α, φi∈£^α )

Ξ^α = span©

̟∈G^⊥∩Ω^α£ such thatιf̟ /∈£^αª .

Letp^α+1= dim Ξ^α. Ifp^α+1= 0, the state of the system (1) can not be recovered with the method described in this paper and the algorithm stops. Otherwise, there exist p^α+1 one-forms ̟i such that Ξ^α = span©

̟1, ., ̟_p^α+1ª

and one can define the following vector of fictitious outputs, suitable to the problem (see Proposition 1): y^α+1 = £

ιf̟1, ..., ιf̟_p^α+1¤T

. Set h

h^α+1₁ , .., h^α+1_pα+1

iT

=

£ιf̟1, ..., ιf̟_p^α+1¤T

. Go to [a].

If the algorithm stops for some µ^k∗ =n, the change of coordinates φ=³

z^k₁^∗, ..., z^k_µ^∗k∗

´T

is well defined and transforms the system into a set of block triangular observable forms similar to (8). The functionshⁱ_j, and the integerspⁱ,lⁱ,ϕⁱ_j,κⁱ_jare obtained in eachi-th iteration of the algorithm.

Remark 2 If the condition (4) is satisfied for the measured outputs of the sys- tem (1), µ¹ = n after the first iteration of the algorithm and the system is exactly transformed into the form (5) that is usually considered for the design of asymptotic (see e.g. [8, 15]) or robust finite time ([6, 22]) nonlinear observers.

4 Estimation of the unknown inputs

If the algorithm ends in a positive way, the unknown inputs can also be obtained in a finite time. Indeed, the use of a finite time observer provides an estimation of the state, say ˜x, and the knowledge of the following quantities (from the last line of each block of the triangular form):

θ˜ⁱ_j =L^κ

i j

f hⁱ_li+j(˜x) +

m

X

s=1

LgsL^κ

i j−1

f hⁱ_li+j(˜x)ws, (9) for 1 ≤ j ≤ pⁱ−lⁱ, and 1 ≤ i ≤ k^∗. The relations (9) can be rewritten as:

Λ(˜x)w= Θ(˜x) with Λ(˜x) =









 L_g

1L^κ

1 1−1 f h¹

l¹+1(˜x) ... L_gmL^κ

1 1−1 f h¹

l¹+1(˜x)

... ...

L_g

1L

κ^k∗ pk∗−l∗−1

f h^k∗

p^k∗(˜x) ... L_gmL

κ^k∗ pk∗−l∗−1

f h^k∗

p^k∗(˜x)











Θ(˜x) =











θ˜¹₁−L^κ

1 1 f h¹

l¹+1(˜x) ... θ˜^k∗

p^k∗−l^∗−L

κ^k∗ pk∗−l∗

f h^k∗

p^k∗(˜x)











Since the distribution span{g1, . . . , gm} is assumed to be nonsingular, the ma- trix

Λ(˜x) =









 dL^κ

1 1−1

f h¹_l1+1(˜x) ... dL^κ

k∗ pk∗−l∗−1 f h^k_pk^∗∗(˜x)











£ g1 · · · gm ¤

has rankmon every subset ofU where at leastm one-formsdL^κ

i j−1

f hⁱ_li+j does not belong to G^⊥. Thus, an estimation of the unknown input is given by:

˜

w= Λ⁺(˜x)Θ(˜x) where Λ⁺is a well defined pseudo-inverse of Λ.

5 Example

As a way of illustration², consider the following nonlinear system subject to the unknown inputw= [w1, w2]^T

˙

x1 = x2−x³₁

˙

x2 = x3+x²₂−x³₂+a(x3, x4)w1

˙

x3 = x5 (10)

˙

x4 = −x4+x²₂+b(x2, x3)w1

˙

x5 = −x3+x2w2

˙

x6 = −x6+w2

with outputsy1=x1,y2=x4andy3=x6. The scalar functionsaandbare such that a(x3, x4) =a1(x3)a2(x4), b(x2, x3) =a1(x3)b2(x2) and wherea(0,0) 6= 0 and b(0,0) 6= 0, and b2(x2) 6= 0, ∀x2 ∈ R. For this system, one has ρ1 = 2, ρ2 =ρ3= 1 and ν1 = 4, ν2 =ν3 = 1 (as a consequenceδ1 = 2, δ2 =δ3 = 1).

Thus, the necessary and sufficient condition (4) for a system to be transformed into a form similar to (5) is not fulfilled. Furthermore, the distributionGis not involutive. As a consequence, the sliding mode observer proposed in [22], where the distribution spanned by the unknown input channels has to be involutive, can not be designed here. However, the procedure proposed in this paper is applicable. The annihilator ofGis given by:

G^⊥= span{dx1, dx3, b2dx2−a2dx4, dx5−x2dx6} In the first step of the algorithm,I¹ is empty and

Ω¹= span{dh1, dLfh1, dh2, dh3}= span{dx1,−3x²₁dx1+dx2, dx4, dx6} with dim Ω¹= 4< n. One has

£¹ = span{h1, Lfh1, h2, h3}= span{x1, x2−x³₁, x4, x6}

= span{x1, x2, x4, x6}

2A physical application of the proposed algorithm in the field of chaotic synchronization for secure communication can be found in [2].

andb2dx2−a2dx4∈Ω¹£. Then G^⊥\

Ω¹£ = span{dx1, b2dx2−a2dx4} Ξ¹ = span{b2dx2−a2dx4}.

Ξ¹6= 0 and one can define the following fictitious output:

y² = ιf(b2dx2−a2dx4)

= b2(x3+x²₂−x³₂)−a2(−x4+x²₂) =b2(x2)x3mod[£]

Sinceb2(x2)6= 0 for allx2, the knowledge ofy² is equivalent to the knowledge of x3. The second step starts by defining the output vector: y² = x3. I² is empty and Ω²= span{dx1, dx2, dx4, dx6, dx3, dx5}with dim Ω²= 6. Then, one can define the change of coordinates: z=ψ(x) = [x1, x2, x4, x6, x3, x5]^T. In the new coordinates, the system is rewritten as:

˙

z1 = z2−z³₁

˙

z2 = z5+z²₂−z₂³+a1(z5)a2(z3)w1

˙

z3 = −z3+z₂²+a1(z5)b2(z2)w1 (11)

˙

z4 = −z4+w2

˙

z5 = z6

˙

z6 = −z5+z2w2

By the means of a finite time observer fed by the original outputs of system (10), i.e. y1=z1,y2=z3andy3=z4, one can recover the state z2 and

V1 = z5+z²₂−z₂³+a1(z5)a2(z3)w1

V2 = −z3+z₂²+a1(z5)b2(z2)w1

V3 = −z4+w2

For the sake of place, the design of the observer is not given here. However, the reader can for instance refer to [7] where an example of higher order sliding mode observer, designed for a similar problem in the linear case, can be extended to the problem of nonlinear systems. Then,

y² = ιf(b2(z2)dz2−a2(z3)dz3)

= b2(z2)¡

z5+z₂²−z₂³+a1(z5)a2(z3)w1¢

−a2(z3)¡

z²₂+a1(z5)b2(z2)w1

¢=b2(z2)V1−a2(z3)V2

= b2(z2)(z5+z₂²−z₂³)−a2(z3)(−z3+z₂²) is known after a finite time andz5= ^y2+a2(z_b^{3)(−z3+z22)}

2(z2) +z³₂−z₂²is an available information. Again, a finite time observer leads to the recovery ofz6. Then, the unknown input can also be obtained since: w1= ^V_a^{1−z5−z22+z32}

1(z5)a2(z3) andw2=V3+z4.

6 Conclusion

In this paper, the problem of the observation of nonlinear systems subject to unknown inputs was considered. An observation algorithm that determines whether it is possible to recover the state and unknown input in finite time was introduced. When the answer is yes, the algorithm provides a change of coordinates that transforms the system in a new type of block triangular observ- able form well suited to the design of finite time observers. The observability matching condition usually required for the design of nonlinear unknown input observers is relaxed with the proposed method.

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